how to??

the last digit is periodic with period 4,

last digits of 7ⁿ is 7,9,3,1,7,9,3,1.........

so, ans is 7

but the ans given is 3...

solution given:

" 7³ ≡ 343 ≡ 3 (mod 10)

(7³)⁶⁷ ≡ (3)⁶⁷ (mod 10)
therefore the last digit is 3 "

oO??

But the reason given by him is also true and the result should come from it.

i agree... but.. that means the solution given in book is either wrong or the answer is indefinite ... ??

The book is 'right' up to ≡3⁶⁷ mod 10. After that to conclude that 3⁶⁷≡3 mod 10 is incorrect.

An easier way using congruences is 7⁴≡1 mod 10 (how did we hit on 4? look up the euler totient function). This is what samagra is saying when he is talking of the last digit being periodic with period 4. The next step elaborates that

So,( 7⁴)⁵⁰ ≡ 7²⁰⁰ ≡ 1 mod 10 and so 7²⁰¹â‰¡7 mod 10.

So the last digit is 7.